Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $r \neq 0$. $q = \dfrac{-10r + 90}{r^2 - 81} \div \dfrac{r + 8}{-2r - 18} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{-10r + 90}{r^2 - 81} \times \dfrac{-2r - 18}{r + 8} $ First factor the quadratic. $q = \dfrac{-10r + 90}{(r + 9)(r - 9)} \times \dfrac{-2r - 18}{r + 8} $ Then factor out any other terms. $q = \dfrac{-10(r - 9)}{(r + 9)(r - 9)} \times \dfrac{-2(r + 9)}{r + 8} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac{ -10(r - 9) \times -2(r + 9) } { (r + 9)(r - 9) \times (r + 8) } $ $q = \dfrac{ 20(r - 9)(r + 9)}{ (r + 9)(r - 9)(r + 8)} $ Notice that $(r - 9)$ and $(r + 9)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac{ 20(r - 9)\cancel{(r + 9)}}{ \cancel{(r + 9)}(r - 9)(r + 8)} $ We are dividing by $r + 9$ , so $r + 9 \neq 0$ Therefore, $r \neq -9$ $q = \dfrac{ 20\cancel{(r - 9)}\cancel{(r + 9)}}{ \cancel{(r + 9)}\cancel{(r - 9)}(r + 8)} $ We are dividing by $r - 9$ , so $r - 9 \neq 0$ Therefore, $r \neq 9$ $q = \dfrac{20}{r + 8} ; \space r \neq -9 ; \space r \neq 9 $